A Variable-Weight Model for Evaluating the Technical Condition of Urban Viaducts

Abstract

Urban viaducts play a crucial role in transportation infrastructure and are closely linked to urban resilience. Accurate evaluation of their structural technical condition forms the basis for the scientific maintenance of urban viaducts. Currently, there is a lack of technical condition evaluation specifications for viaducts in China, and the existing bridge specifications that are similar do not fully align with the facility composition characteristics and maintenance management needs of viaducts. Therefore, this paper presents a technical condition assessment model for viaducts, based on existing bridge specifications. Considering the frequent damage to ancillary facilities of viaducts, the utilization of maintenance resources, and the impact on traffic operations, the model proposed in this paper adopts the Analytic Hierarchy Process (AHP) to introduce a new indicator layer for ancillary facilities. Subsequently, the weight values and deduction values of each layer of the model, as well as the findings of damage recorded in the new components, were determined using the Group Decision-Making (GDM) method and the Delphi method. This process forms a constant-weight evaluation model for assessing the technical condition of viaducts. Finally, to account for the impacts of significant damage to low-weight components on the structural condition, the variable-weight method was adopted to establish a comprehensive evaluation model with variable weights, which was then validated using practical viaduct examples. The results indicate that the variable-weight model provides a more accurate representation of the technical condition of viaducts, especially when components are severely damaged. Furthermore, this study examines the suitable conditions for implementing the constant-weight evaluation model and the variable-weight evaluation model, demonstrating that the variable-weight model is recommended when there is a significant disparity in the scores among the viaduct components, whereas the constant-weight model is applicable in other scenarios.

1. Introduction

An increasing number of scholars have explored the concept and characteristics of urban resilience, as documented in the existing literature [1,2]. Research from various disciplines and perspectives has provided definitions for urban resilience [3,4,5,6]. Currently, there is a general belief that urban resilience is defined as the ability of a city to maintain its main features, structure, and key functions in the face of external disturbances [7,8]. A comprehensive risk assessment of urban resilience can provide early predictions and subsequent decision-making information for urban policymakers, which is crucial for their decision-making processes [9]. This process contributes to improving overall economic development, public safety, and government management, exerting a significant impact on the progress towards urban sustainability goals [10,11,12]. There exists a strong correlation between infrastructure and urban resilience, and maintaining the well-being of infrastructure is essential for the effective operation of most municipal systems [13]. Specifically, the development of urban transportation infrastructure ensures the city’s robust recovery and reconstruction capabilities from the perspective of urban safety [14,15]. The integrity of transportation infrastructure not only guarantees the capacity of transportation services but also positively influences the achievement of urban sustainability goals [16,17].

The current focus of the resilience concept in transportation infrastructure management lies primarily in ensuring facility safety. This includes activities such as safety assessment [18], safety-risk assessment [19,20], and safety-risk prediction [21,22,23]. Scholars have extensively examined and analyzed the safety-risk factors in various road transportation infrastructures, mainly for bridges [24] and tunnels [25]. For instance, Wei et al. [26] proposed a method utilizing multiple types of monitoring data to predict the service life of sliding bearings on bridges, demonstrating an effectiveness in estimating cumulative displacement. Liu et al. [27] conducted an analysis of sensitivity indicators for tunnel monitoring parameters, quantifying and ranking the sensitivity of parameters. They identified sensitive parameters (opening width of the pipe layer, longitudinal settlement value of the tunnel, and contact stress between tunnel segments) and insensitive parameters (soil temperature and concrete corrosion depth). Zhao [28] proposed a reliability-based optimization design method for support structures by combining the moment method with High-Dimensional Model Representation (HDMR). While current research predominantly concentrates on urban tunnels and bridges within the transportation infrastructure, there is limited research on the structural safety assessment of urban viaducts. A viaduct is a type of spanning bridge, generally referred to as an overhead bridge, over a series of reinforced concrete or masonry arches with supporting columns underneath, and spanning low-level obstacles such as roadways.

In large cities, viaducts form a critical component of the expressway network, serving as the core of the urban transportation system, with stringent safety and security requirements. Ensuring the safety of viaducts is a crucial aspect of urban resilience assessment. However, due to the relatively recent construction of urban viaducts, there has been limited research on their structural issues and associated challenges. In recent years, there has been a growing incidence of damage in early-period viaducts, highlighting the urgency of conducting safety evaluations for such structures. However, a mature model for the safety evaluation of urban viaducts has yet to be developed. Therefore, this study aims to investigate the development of a structural safety evaluation model for urban viaducts, with the objective of enhancing the understanding of the relationship between transportation infrastructure and urban resilience.

2. Literature Review

2.1. Evaluation Methods for Small and Medium-Sized Bridges

Urban viaducts are structurally similar to ordinary small and medium-sized bridges, and therefore, their structural evaluation can be based on the evaluation methods used for small and medium-sized bridges. The technical condition of bridges serves as the fundamental basis for evaluating the structural safety, maintenance urgency, and operational service level. The objectivity, comprehensiveness, accuracy, and applicability of the assessment results directly impact the effectiveness of maintenance management [29], while the evaluation is based on the inspection data. Therefore, it is crucial to use suitable methods for inspection and evaluation to ensure effective bridge maintenance. However, currently, there is no globally unified and authoritative standard for data collection and technical condition assessment. Only certain countries, such as the United States [30], the United Kingdom [31], Germany [32], Japan [33], China [34], etc., have released qualitative management guidelines that provide recommendations on management principles, data collection cycles, precautions, and so on. Therefore, the evaluation method for viaducts studied in this paper is primarily based on the bridge evaluation methods of the above countries.

Currently, the bridge evaluation methods employed in the United States primarily utilize the Bridge Health Index (BHI) [35] as a fundamental indicator for assessment. Scholars have conducted extensive research on the application of BHI in bridge management in California [36,37]. Additionally, they have attempted to use the Analytic Hierarchy Process (AHP) to determine the BHI [38] and explored weight calculation methods based on availability [39]. In the United Kingdom, the primary method employed is the weighted-average approach [40], assigning weights based on the importance of components. The Condition Performance Indicator (PI) [41] is utilized as an indicator to assess the technical condition of bridges. Both Germany and Japan have adopted the ‘worst component’ evaluation method [42,43], in which the ‘worst component’ refers to the component with the most severe damage. The difference between the two is that in Japan, the components with the highest defect level among all structural parts are combined, while in Germany, the bridge condition index is directly derived from the state of the worst component. China adopts a layered weighted method, utilizing the Bridge Condition Index (BCI) as the indicator [44]. This study compared the evaluation methods from various countries in terms of evaluation processes, hierarchical levels, weight distribution, etc. The comparison, as presented in

Table A1. Comparison of methods for evaluating the technical condition of bridges.

Country	Evaluation Method and Indicators	Evaluation Process	Levels of Evaluation	Weight Distribution	Damage Level
United States	Bridge Health Index (BHI)	Component Failure Cost (FC)–Component Remaining Value (CEV)–Bridge Health Index (BHI)	Component–Whole Bridge	Fixed Weights	5 levels
United Kingdom	Condition Performance Indicator (PI); Average Structural Condition Score (SCSAv) and Key Structural Condition Score (SCSCrit)	Severity of Damage—Element Condition Score (ECS)–Element Condition Factor (ECF)–Element Condition Index (ECI)–Average Structural Condition Score (SCSAv) and Key Structural Condition Score (SCSCrit)–Condition Performance Indicator (PI)	Damage-Component–Whole Bridge	Fixed Weights	5 levels
Germany	The ‘Worst Component’ Evaluation Method	Damage Index (Zi)–Component Group Condition Index (ZCGi)–Overall Structural Condition Index (Zges)	Damage-Component–Component Group–Whole Bridge	Fixed Weights	5 levels
Japan	The ‘Worst Component’ Evaluation Method	Damage Grade–Component Deficiency Level (d)–Component Deficiency Level–Bridge Technical Condition Score	Damage-Component–Part–Whole Bridge	Fixed Weights	5 levels
China	Hierarchical Weighting Method; Bridge Condition Index (BCI) and Bridge Structural Index (BSI)	Specific Component Damage–Component Evaluation–Overall Bridge Evaluation	Damage-Component–Part–Whole Bridge	Fixed Weights	3 levels

(Appendix A), serves as a reference for selecting the method of evaluating the technical condition of viaducts.

2.2. Difference between Urban Viaducts and Ordinary Bridges

Although urban viaducts are structurally similar to ordinary small and medium-sized urban bridges, they still have several differences. The main challenge stems from the diverse range of auxiliary facilities associated with urban viaducts. Consequently, when employing existing bridge evaluation methods to assess viaduct structures, certain components’ actual damage types may not align with the current specifications, rendering this portion of data unavailable. This, in turn, leads to an overestimation of the technical condition of the viaducts and results in some degree of data resource wastage.

In China, the ‘Technical standard of maintenance for city bridge’ (CJJ99-2017) [45] is the most suitable standard for urban viaducts.

Table 1. Comparisons of viaduct components with bridge components described in the current Chinese standard.

Structural Parts	Urban Bridge Components in Current Standard	Urban Viaduct Components
Superstructure	Steel structure beam PC or RC girder Cross beam	Steel structure beam PC or RC girder Cross beam
Substructure	Cover girder Pier body Pier cap Pier shaft Bearing Foundation Abutment wing wall	Cover girder Pier body Bearing Foundation
Bridge deck system	Bridge deck pavement Drainage system Pedestrian walkway Expansion joint Railings or guardrail Bridge head smoothing	Bridge deck pavement Drainage system Expansion joint Railings or guardrail Bridge head smoothing
Ancillary facilities	None	Sound barrier Anti-glare screen Anti-throwing net (Forbidden grid) Gantry (TF pole) Inducer Anti-collision bucket Isolation pier

provides a comparison between the corresponding components of urban bridges and viaducts.

From Table 1, it can be determined that the primary distinction between urban bridges and viaducts lies in their ancillary facilities. Ancillary facilities are not classified as a separate category in the current classification of urban bridge components. The bridge deck system incorporates a few ancillary facilities, including drainage systems and railing barriers. However, the bridge technical condition evaluation model assigns a very low weight to these ancillary facility components. According to the current Chinese standard [45], in the case of a beam bridge, the drainage system has a weight of 10% in the bridge deck system score, whereas the bridge deck system accounts for 15% in the overall bridge score. Consequently, the drainage system only contributes 1.5% to the overall bridge score. In the case of small to medium-sized urban bridges in general, it is justifiable for ancillary facilities to account for a relatively low proportion in the bridge score, considering their minimal impact on the overall structural safety of the bridge.

Presently, the fundamental objective of bridge management is to ensure regulated structural safety, and defects in ancillary facilities typically do not have a direct impact on the traffic service capacity of the bridge [46]. However, this scenario does not apply to urban viaducts. During the service life of urban viaducts, ancillary facilities are prone to frequent damage, which can directly impact the traffic service performance of the viaducts. Therefore, it is imperative to adequately consider ancillary facilities in the assessment of the technical condition of urban viaducts.

Furthermore, while the current evaluation methods in Chinese standards offer a relatively comprehensive understanding of the overall condition of urban viaducts, especially in terms of safety, they do not provide sufficient guidance for bridge maintenance, specifically in terms of daily maintenance. This is because damage commonly occurs in the bridge deck system and ancillary facilities, which have low component weights and deduction values. Consequently, adequately reflecting these incidents of damage in the Bridge Condition Index (BCI) poses a challenge. Hence, it is essential to integrate ancillary facility indicators into the current technical condition evaluation methods and incorporate variable weighting theory [47,48], a practice which can effectively consider the balance and discrepancies between indicators [49].

Therefore, a variable-weight evaluation model for urban viaducts has been proposed in this study. Firstly, the regular inspection data and daily maintenance data of a viaduct was collected and a new evaluation framework for viaducts was established, following the logic of the current standard [45]. Then the weight of each component was determined by the Analytic Hierarchy Process (AHP) and Group Decision-Making (GDM) methods. As for the newly added ancillary facilities components, their damages were defined on the basis of an engineering survey, and the Delphi method was applied to grade their damage status as well as to set the deduction value. As a result, a constant-weight evaluation model for urban viaducts was obtained. Finally, considering that components with small weights may also lead to severe consequences when the damage is significant, this paper introduced the variable weighting theory to refine the weight allocation method.

3. Methodology

The research idea of this paper is depicted in Figure 1. Initially, the maintenance data of viaducts is collected, followed by the conducting of essential data processing. Subsequently, the evaluation of ancillary facilities is introduced based on the existing evaluation standards of urban bridges [45] to establish a framework for assessing the technical condition of viaducts which is tailored to their specific requirements. The framework distinctly assesses the content and scope of viaduct evaluation. Subsequently, determining the weights of the evaluation indices and the impact of component damage becomes essential. This study sequentially employs methods such as the AHP, GDM and Delphi methods, among others, to derive a constant-weight model for viaduct assessment. Recognizing that the significance of severe damage to individual components may diminish with multi-layer weighting, this study employs a variable-weight method to enhance the constant-weight model, resulting in the variable-weight model for viaduct evaluation.

3.1. Data Preparation

The viaduct investigated in this study has a main line that spans 28.11 km and consists of 1307 spans, mainly in the form of continuous girders of box-type concrete structures. The data collected for this study include regular inspection data and daily maintenance data, spanning from 2012 to 2019. The regular inspection data primarily exists in the form of report text, which provides a more systematic overview, but contains fewer specific details in the data. The daily maintenance data are the largest and most widely distributed, but the data is also relatively scattered and requires sorting and cleaning before utilization. Prior to analyzing the data, this study conducted data preprocessing on the daily maintenance data and filtered out valid data with complete information such as reporting time, damage type, and damage quantity. Following the sorting process, a total of 41,585 pieces of daily maintenance data were obtained.

The collected data were analyzed statistically, as presented in

Table 2. Damage to components of the investigated viaducts (2012–2019).

Items	Superstructure	Substructure	Deck System and Ancillary Facilities
Number of damage incidents (or frequency)	893	190	40,502
Number of damage types	8	13	64

, based on the number of damaged components and types of damage in different parts of the bridge. It can be determined that the bridge deck system and ancillary facilities of viaducts exhibit a significantly higher number of damaged components and damage types compared to superstructures and substructures, highlighting the high frequency of damage incidents in the daily operation of the former. Thus, emphasis should be placed on the bridge deck system and ancillary facilities.

3.2. Component Framework of the Viaduct Evaluation Model

Based on the collected data above, a statistical analysis of the component types and damage frequency in the deck system and ancillary facilities was conducted. The components were also analyzed in comparison to the current standard [45], as shown in

Table 3. Comparison of viaduct deck system and ancillary facilities with those in the current standard.

Viaduct Components	Damage Frequency	Corresponding Component in Current Standard	Structural Parts of The Viaduct
Bridge deck pavement	10,834	Bridge deck pavement	Bridge deck system
Railings or guardrails	563	Railings or guardrails	Bridge deck system
Crash barrier	3456	Railings or guardrails	Bridge deck system
Cross drain	3611	Drainage system	Bridge deck system
Inlet	5182	Drainage system	Bridge deck system
Drainage pipe	785	Drainage system	Bridge deck system
Expansion joint	1142	Expansion device	Bridge deck system
Sound barrier	8131	None	Ancillary facilities
Anti-glare screen	3448	None	Ancillary facilities
Gantry (TF pole)	334	None	Ancillary facilities
Anti-throwing net (Forbidden grid)	977	None	Ancillary facilities
Isolation pier	674	None	Ancillary facilities
Inducer	185	None	Ancillary facilities
Anti-collision bucket	1180	None	Ancillary facilities

.

For components already included in the current standard [45], the rating framework can be utilized, but the definition of damage needs to be revised to align with the characteristics of the viaduct. As for components not included in the current standard, they are classified as ancillary facility components with a rating tier equivalent to that of the bridge deck system.

This study classified the viaduct structure into three levels: components, structural parts, and the whole viaduct. The evaluation is carried out sequentially, and the overall technical condition score of the viaduct is obtained through hierarchical weighting. The three levels are identified as A, B, and C, respectively. “A” corresponds to the overall technical condition evaluation of the whole viaduct; “B” corresponds to the condition evaluation of four parts, namely, the superstructure, substructure, deck system, and ancillary facilities; and “C” corresponds to the component condition evaluation of each part. Specific details can be found in Figure 2.

Compared with the current standard [45], the improved technical condition evaluation model framework for viaducts has incorporated ancillary facilities within the part level “B”, which encompasses seven types of components. Moreover, considering the structural characteristics of the viaducts, the evaluation model excludes pedestrian walkways from the deck system, and omits bridge pier components from the substructure.

3.3. Component Weight Calculation

After developing the component framework for the viaduct evaluation model, it is necessary to further determine the weights assigned to each component. The primary methods utilized in this paper are the Analytic Hierarchy Process (AHP) method and the Group Decision-Making (GDM) method.

AHP is a comprehensive evaluation method designed to address multi-criteria problems and the quantification of these criteria [50]. It has the capability to represent complex problems as ordered hierarchical structures and employs judgments from decision-makers to rank the superiority or inferiority of decision alternatives. This method is widely adopted in various fields, such as engineering management, economic analysis, risk assessment, etc. [51,52,53,54,55]. Generally, the process of establishing decision criteria weights through AHP can be completed in three steps [56,57]. Firstly, the decision hierarchy structure needs to be formulated and the indicator system needs to be built. Next, the indicators are compared in pairs, and a comparison matrix of each indicator is constructed based on the importance level, which is provided by experts. The consistency test is conducted to ensure the robustness of the weighted system. Finally, the weights for each indicator are calculated using the provided comparison matrices. The flowchart shown in Figure 3 summarizes the entire process of AHP method implementation.

The establishment of a weight model using the AHP method relies on expert experiential judgment. However, when it comes to complex bridge systems, relying solely on the judgment of one expert can easily result in distortion or even misjudgment. Therefore, the incorporation of judgments from multiple experts is necessary. In addition, the level of expert judgment can be represented by weights, and the use of weighted averages will help to improve the accuracy of expert judgments. The judgment matrix filled in by the experts actually indirectly reflects their professional level. Therefore, by employing specific methods, the reliability of judgment matrices can be assessed, and this assessment can be used to determine the weights of the experts. This approach is commonly referred to as GDM, which is often combined with AHP and widely utilized in solving multi-attribute decision-making problems [58,59,60,61,62].

Therefore, the expert judgment matrices can be obtained first by using the AHP method. Subsequently, the credibility of the judgment matrix is assessed, with the help of GDM, based on two factors: (1) dissimilarity with other experts and (2) similarity with the comprehensive results. Finally, the credibility of the judgment matrix is calculated based on dissimilarity and similarity, and this value represents the weight of the expert. The specific steps of the GDM method are as follows:

(1)

Calculation of variability in GDM

The variability of the experts’ judgments is reflected by the standard deviation. Suppose there are x experts and accordingly, x judgment weight column vectors $[{\delta }_1,{\delta }_2,\dots ,{\delta }_x]$ are generated, where ${\delta }_i=[{\delta }_{k1},{\delta }_{k2},\dots ,{\delta }_{kn}]$ denotes the value of the judgment made by the kth expert on the n judgment indicators. Let ${\sigma }_{ki}$ be the standard deviation of the judgment made by the kth expert on the ith judgment indicator, where k represents the expert code and i represents the evaluation indicator. From the definition of standard deviation, Equation (1) can be obtained.

$${\sigma }_{ki}=\left|{\delta }_{ki}-\frac{1}{x}{\sum }_{k=1}^x{\delta }_{ki}\right|,k=1,2,\dots ,x,i=1,2,\dots ,n$$

(1)

Let ${\sigma }_k$ denote the sum of the kth expert’s standard deviation values for each judgmental indicator, and ${\sigma }_k$ is given in the following equation:

$${\sigma }_k=\sum _{i=1}^n{\sigma }_{ki}$$

(2)

The coefficient of variation ${\lambda }_k$ is obtained by normalizing ${\sigma }_k$, and ${\lambda }_k$ is defined in the following equation:

$${\lambda }_k=\frac{{\sigma }_k}{{\sum }_{k=1}^x{\sigma }_k}$$

(3)

(2)

Similarity calculation in GDM

The similarity coefficient is calculated by using the spatial location relationship between the vectors as a reflection of the expert judgment similarity index, i.e., the magnitude of the cosine of the angle between the two vectors is used to represent the similarity.

The judgment weights obtained by AHP are column vectors, and there are 2 column vectors $\alpha$ and $\beta$, where $\alpha ={[a_1,a_2,\dots ,a_n]}^T$ and $\beta ={[b_1,b_2,\dots ,b_n]}^T$, and the angle of the vectors $\alpha$ and $\beta$ is $\theta$, which is defined by the cosine in the following equation:

$$\mathit{cos}\theta =\frac{⟨\alpha ,\beta ⟩}{‖\alpha ‖‖\beta ‖}=\frac{{\sum }_{i=1}^n{a}_i{b}_i}{\sqrt{{\sum }_{i=1}^n{a_i}^2}\sqrt{{\sum }_{i=1}^n{b_i}^2}}$$

(4)

where $\eta =\mathit{cos}\theta ,0\le \eta \le 1$, if and only if $\alpha =k\beta ,\eta =1$.

Suppose there are x experts, corresponding to x column vectors of judgment weights $[{\delta }_1,{\delta }_2,\dots ,{\delta }_x]$. Let ${\eta }_{ij}$ be the cosine of ${\delta }_i,{\delta }_j$; then, ${\eta }_{ij}$ is defined by the following equation:

$${\eta }_{ij}=\mathit{cos}\theta =\frac{⟨{\delta }_i,{\delta }_j⟩}{‖{\delta }_i‖‖{\delta }_j‖}$$

(5)

Let ${\eta }_i$ denote the sum of similarity between expert i and other experts, as shown in Equation (6). The larger ${\eta }_i$ is, the higher its similarity with other experts’ judgment results, and therefore the higher the credibility of its judgment weight column vector, and, vice versa, the lower its credibility.

$${\eta }_i=\sum _{\mathrm{j}=1}^m{\eta }_{ij}-1$$

(6)

Normalizing ${\eta }_i$ yields the similarity coefficient ${\mu }_i$ as depicted in the following equation:

$${\mu }_i=\frac{{\eta }_i}{{\sum }_{j=1}^m{\eta }_i},i=1,2,\dots ,x$$

(7)

(3)

Calculation of credibility in GDM

The higher the credibility coefficient, the closer the expert’s judgment is to the comprehensive result of the group decision. ${\lambda }_k$ and ${\mu }_k$ are the difference coefficient and similarity coefficient of expert k. Let the credibility coefficient of expert k be ${\omega }_k$, as shown in Equation (8).

$${\omega }_k=\left\{\begin{array}{cc}\frac{{\mu }_k\left(1-{\lambda }_k\right)}{1-{\sum }_{k=1}^x{\mu }_k{\lambda }_k}& {\displaystyle \sum _{k=1}^x}{\mu }_k{\lambda }_k\ne 1\\ {\mu }_k& {\displaystyle \sum _{k=1}^x}{\mu }_k{\lambda }_k=1\end{array}\right.$$

(8)

Combining the above methods, this study derived the weights of various components in the viaduct evaluation model from the scoring results provided by five senior experts. These weights are presented in

Table 4. Weights of various components in the viaduct evaluation model.

Structural Parts	Part Level	Part Weight	Viaduct Components	Component Level	Component Weight
Ancillary facilities	B1	0.10	Anti-collision bucket	C1	0.03
			Inducer	C2	0.05
			Anti-glare screen	C3	0.14
			Isolation pier	C4	0.09
			Anti-throwing net (Forbidden grid)	C5	0.21
			Gantry (TF pole)	C6	0.23
			Sound barrier	C7	0.25
Bridge deck system	B2	0.16	Bridge deck pavement	C8	0.33
			Bridge head smoothing	C9	0.17
			Expansion joint	C10	0.28
			Drainage system	C11	0.11
			Railings or guardrails	C12	0.11
Superstructure	B3	0.34	Main girder	C13	0.60
			Cross beam	C14	0.40
Substructure	B4	0.40	Cover girder	C15	0.15
			Pier body	C16	0.30
			Foundation	C17	0.40
			Bearing	C18	0.15

.

3.4. Determination of Condition Ratings of Viaduct Components

After establishing the evaluation framework for assessing the technical condition of viaducts and determining the scoring weights for individual components, the subsequent task was to tackle the calculation of scores for each individual component. This study utilized a layered weighting approach based on deduction points. According to this method, the damage type of each component (component level) should be determined initially, and points should be deducted for each type of damage, based on its severity according to the current standard [45]. The deductions are then weighted to obtain a score for that component. Subsequently, once the scores for each component are acquired sequentially, the components are weighted and summed to derive the score for the upper level. This process continues iteratively, with each level being weighted, until the final score for the viaduct is obtained. Thus, it is crucial to provide clear definitions of component damage, establish a grading system for damage, and determine the appropriate deduction values for each grade.

3.4.1. Definition of Component Damage in Ancillary Facilities

Compared to the current standard [45], the proposed evaluation framework for viaducts adds a new category in the parts layer, specifically focusing on ancillary facilities (Figure 2). The rating definitions for other components remain consistent with the current specifications. Hence, the emphasis is on defining the damage incurred by components associated with ancillary facilities, as illustrated in

Table 5. Damage types for each component of the ancillary facilities.

Components	Damage Type	Damage Definition
Sound barrier	Glass shattering	Screen glass shattering or detachment
	Corrosion of steel structure	Surface corrosion of steel structure
	Cover plate detachment or loss	The upper or lower cover plate is detached and defective
	Screen panel damage	Damage to the outer frame of the screen panel; scratches or defects on the surface of pc panel
	Bolt and nut deficiency	Loosening or fracture of anchor bolts supporting columns; missing nuts; displacement of base gaskets
	Loose or misaligned	Tilt or displacement of screen body in sound barrier
Anti-glare screen	Screen panel damage	Structural damage to the screen panel
	Corrosion of steel structure	Surface corrosion of steel structure
	Loose or misaligned	Tilt, loosening, or misalignment of the screen body
		Loosening or misalignment of the base
	Bolt and nut deficiency	Loosening or fracture of anchor bolts, missing nuts
Gantry (TF pole)	Damage to the height restriction pole	Overturning, deformation, and deficiency of the height restriction pole
	Corrosion of steel structure	Surface corrosion of steel structure
	Peeling of concrete foundation	Concrete cracking and spalling on the surface of concrete structure
	Deformation or damage of steel pipe	Deformation or damage of the steel pipe structure
	Bolt and nut deficiency	Rust on screws; loosening or fracture of bolts; missing nuts
Anti-throwing net (Forbidden grid)	Corrosion of steel structure	Surface corrosion of steel structure
	Mesh damage	Mesh deficiency
	Loose or misaligned	Column tilting and loosening
	Bolt and nut deficiency	Rust on screws; loosening or fracture of bolts; missing nuts
Isolation pier	Damage to reflective column	Column deficiency and deformation
	Coating peeling	Paint peeling; coating peeling
	Isolation pier displacement	Isolation pier displacement
	Isolation pier concrete peeling	Isolation pier surface concrete cracking and peeling
Anti-collision bucket	Barrel head deficiency	Missing anti-collision bucket lid
	Misalignment	Misalignment of anti-collision bucket
	Barrel body deficiency	Bucket body damage
	Retroreflective film damage	Surface retroreflective film deficiency on anti-collision bucket
Inducer	Deficiency	Damage or loss of inducer

.

3.4.2. Determination of Demerit Points for Component Damage in Ancillary Facilities

After clarifying the definition of component damage, it is necessary to further classify the levels of damage and establish deduction points. This study utilizes the Delphi method. The Delphi method is a rigorously designed research technique employed to acquire independent expert opinions on specific topics [63]. The Delphi technique is more accurate than other traditional survey methods because it enables researchers to control biased responses from qualified experts. Based on this, the controlled responses obtained in multiple rounds can efficiently aid experts in reaching a consensus [64,65]. Its primary features include anonymity and avoidance of face-to-face contact, which can prevent groupthink and enhance confidentiality [66]. However, the Delphi method has certain drawbacks; for example, conducting multiple rounds of surveys with experts can be time-consuming, often resulting in participant attrition and limitations [67]. In summary, the Delphi method was chosen for this study due to the following reasons: (1) It has been extensively utilized in the research of evaluating the structural condition of bridges. (2) The anonymous nature of the method provides participants with the freedom to express their judgments, resulting in more authentic responses. (3) By collecting expert opinions through repeated questionnaire surveys, the Delphi method enhances the representativeness and reliability of the data.

The damage severity classification for each component was obtained through consultation with experienced experts. Based on the Delphi method, a score of 1 to 9 points was assigned to each damage classification to represent the impact level associated with the component (1 for least important, 9 for most important). Five experts (senior technicians who have been engaged in bridge maintenance for more than 10 years) were invited to provide scores, and their scoring results were examined. The allowable threshold for differences in expert scores was set at 2 points. If the maximum score difference among different experts exceeded 2 points, the current round of scoring was considered invalid. This then required a reevaluation and discussion with the experts regarding the definitions of component damage and the impact classifications. The process was repeated until all component damage severity scores passed the consistency check.

The damage deduction range for each component obtained based on the Delphi method ranges from 1 to 9 points. However, according to the current Chinese standard, the deduction range for component damage is from 0 to 80 points [45], with graded deduction points in multiples of 5 points. Therefore, in order to align with the existing specifications, the deduction points obtained from the Delphi method need to be converted, as shown in

Table 6. Correspondence between the average score of experts and the final deduction points.

Average Expert Score	Deduction Range for Corresponding Specification	Median Value of Deductions	Converted Deduction Points
0	0	0	0
1~2	80/9~160/9	120/9	15
2~3	160/9~240/9	200/9	20
3~4	240/9~320/9	280/9	30
4~5	320/9~400/9	360/9	40
5~6	400/9~480/9	440/9	50
6~7	480/9~560/9	520/9	60
7~8	560/9~640/9	600/9	65
8~9	640/9~720/9	680/9	75
9	80	80	80

.

In this way, the grading and deduction points for all components of the viaduct ancillary facilities are obtained, as shown in

Table A2. Scoring levels and deduction points (DP) for each component of ancillary facilities.

Component	Damage Type	Scoring Levels				Explanation
Sound barrier	Glass shattering	Grade	None	≤5	>5	Number of sound barriers with broken glass.
		DP	0	60	75
	Corrosion of steel structure	Grade	None	≤10%	>10%	The percentage of total area of steel structure affected by corrosion.
		DP	0	25	45
	Cover plate detachment or loss	Grade	None	≤5	>5	The quantity of detached cover panels.
		DP	0	30	40
	Screen panel damage	Grade	None	Slight	Severe	“None” means no damage; “Slight” refers to minor damage to the outer frame of the sound barrier that does not affect structural functionality; “Severe” indicates significant damage to the outer frame that affects structural functionality.
		DP	0	30	60
	Bolt and nut deficiency	Grade	None	≤5	>5	Number of loose, broken, or missing bolts and nuts.
		DP	0	15	40
	Loose or misaligned	Grade	Slight	Medium	Severe	“Slight” means that individual components of the screen body are loose and misaligned and this does not affect safety; “Medium” means that some components of the screen body are loose and misaligned, affecting the structural function; “Severe” means that a large number of components are seriously loose and misaligned, seriously affecting the structural function.
		DP	10	30	80
Anti-glare screen	Screen panel damage	Grade	None	Slight	Severe	“None” means no damage; “Slight” means that the outer frame of the anti-glare screen is defective but this does not affect the structural function; “Severe” means that the outer frame of the anti-glare screen is obviously defective and this affects the structural function.
		DP	0	15	30
	Corrosion of steel structure	Grade	None	≤10%	>10%	The percentage of total area of steel structure affected by corrosion.
		DP	0	25	45
	Bolt and nut deficiency	Grade	None	≤5	>5	Number of loose, broken, or missing bolts and nuts.
		DP	0	15	40
	Loose or misaligned	Grade	Slight	Medium	Severe	“Slight” means that individual components of the screen body are loose and misaligned, but this does not affect safety; “Medium” means that some components of the screen body are loose and misaligned, affecting the structural function; “Severe” means that a large number of components are seriously loose and misaligned, seriously affecting the structural function.
		DP	10	30	80
Gantry (TF pole)	Damage to the height restriction pole	Grade	None	Slight	Severe	“None” means no damage; “Slight” means that the height restriction bar is defective, but this does not affect the structural function; “Severe” means that the height restriction bar is obviously defective, affecting the structural function.
		DP	0	30	60
	Corrosion of steel structure	Grade	None	≤10%	>10%	The percentage of total area of the steel structure affected by corrosion.
		DP	0	25	45
	Bolt and nut deficiency	Grade	None	≤5	>5	Number of loose, broken, or missing bolts and nuts.
		DP	0	15	40
	Peeling of concrete foundation	Grade	None	Slight	Severe	“None” means no concrete spalling; “Slight” means that concrete spalling exists but is not obvious; “Severe” means that concrete spalling is obvious and affects function.
		DP	0	30	45
	Deformation and damage of steel pipe	Grade	Slight	Medium	Severe	“Slight” means that the structural deformation of the steel pipe is not obvious; “Medium” means that the steel pipe is deformed but this does not affect the structural function; “Severe” means that the steel pipe is seriously deformed and damaged and that this affects the structural function.
		DP	15	40	80
Anti-throwing net (Forbidden grid)	Corrosion of steel structure	Grade	None	≤10%	>10%	The percentage of total area of the steel structure affected by corrosion.
		DP	0	25	45
	Mesh damage	Grade	Slight	Medium	Severe	“Slight” means that the surface of the mesh begins to appear defective, only slightly affecting aesthetics, but this does not affect the structural function; “Medium” means that the mesh is defective in the area of ≤ 20%, not only affecting aesthetics, but also a certain potential safety hazard; “Severe” refers to defects in more than 20 per cent of the area of the mesh, and not only does this seriously affect the aesthetics, there are also serious security risks.
		DP	20	30	50
	Bolt and nut deficiency	Grade	None	≤5	>5	Number of loose, broken, or missing bolts and nuts.
		DP	0	15	40
	Loose or misaligned	Grade	Slight	Medium	Severe	“Slight” means that individual components of the anti-throwing net are loose and misaligned, but this does not affect safety; “Medium” means that part of the components of the anti-throwing net are loose and misaligned, affecting the structural function; “Severe” means that a large number of components are seriously loose and misaligned and this seriously affects the structural function.
		DP	10	30	80
Isolation pier	Damage to reflective column	Grade	None	Slight	Severe	“None” means no damage; “Slight” means that the piers have defects, but this does not affect the structural function; “Severe” means that the piers have obvious defects, affecting the structural function.
		DP	0	20	40
	Coating peeling	Grade	None	≤10%	>10%	Percentage of the total area of peeling coating and paint flaking over the entire surface area of the steel structure.
		DP	0	20	40
	Loose or misaligned	Grade	Slight	Medium	Severe	“Slight” refers to pier body individual component loose dislocation which does not affect safety; “Medium” refers to pier body part component loose dislocation which affects the structural function; “Severe” refers to a large number of components with severe loose dislocation, seriously affecting the structural function.
		DP	10	30	80
	Isolation pier concrete peeling	Grade	None	Slight	Severe	“None” means no concrete spalling; “Slight” means concrete spalling exists but is not obvious; “Severe” means the concrete spalling is obvious and affects its function.
		DP	0	30	45
Anti-collision bucket	Barrel head deficiency	Grade	None	Slight	Severe	“None” means that there are no defects in the lids of the crash casks; “Slight” means that there are individual crash casks with lids missing, which poses a certain degree of safety hazard; “Severe” means that there are a large number of crash casks with lids missing, which poses a serious safety hazard.
		DP	0	20	40
	Barrel body deficiency	Grade	Slight	Medium	Severe	“Slight” means that the barrel body of the anti-collision bucket is slightly damaged due to scratches, which only slightly affects the aesthetics but does not affect the structural function; “Medium” means that the barrel body of the anti-collision bucket is obviously damaged and partially deformed by the vehicle impact, which not only affects the aesthetics, but also poses a certain security risk; “Severe” means that the anti-collision bucket is seriously flattened or damaged, which not only seriously affects the aesthetics, but also poses a serious security risk.
		DP	20	30	50
	Retroreflective film damage	Grade	None	Slight	Severe	“None” means that the reflective film is not damaged; “Slight” means that the surface of the reflective film is slightly damaged due to scratches, which only slightly affects the aesthetics but does not affect the structural function; “Severe” means that the surface of the reflective film is seriously damaged, which not only seriously affects the aesthetics, but also poses a serious potential safety hazard.
		DP	0	15	30
	Misalignment	Grade	Slight	Medium	Severe	“Slight” means that the collision avoidance drum starts to appear loose and misaligned, which does not affect safety; “Medium” means that the collision avoidance drum is more obviously misaligned, which affects the structural function, and there is a certain potential safety hazard; “Severe” means that the crash cushion is seriously misaligned, seriously affecting the structural function and posing a serious safety hazard.
		DP	10	30	80
Inducer	Deficiency	Grade	None	≤5	>5	Number of defective inducers.
		DP	0	15	30

(Appendix A). The BCI is expressed as a percentage index according to the current standard [45]. Each component is initially assigned a full score of 100 points, which is then reduced based on the extent of damage, following a deduction system that correlates with the degree of damage. The final score for the whole bridge is determined by applying hierarchical weights to its individual components. Following this approach, the scores of ancillary facility components of the viaduct can be derived using Table A2 (Appendix A), and subsequently, the total score of the viaduct can be obtained based on the weights listed in Table 4.

3.5. Adjustment of Weights Based on the Variable-Weights Method

It can be observed from Table 4 that the weight assigned to ancillary facilities is only 0.1. Nonetheless, damages to ancillary facilities’ components can considerably impact the daily maintenance costs of viaducts and elevate the risk of traffic operations. Consequently, there is a need to increase the focus on the components of ancillary facilities when evaluating the technical condition of the viaducts. To address this, the variable-weight method was used to refine the weights assigned to different components of the viaducts.

The variable-weight method is a dynamic modeling method [68] which is used to adjust weights, addressing situations in which constant weights may lead to decision results biased towards extremes. This method is widely applied in decision-making domains that emphasize the variation of indicator weights [69]. The basic principles are outlined as follows [70]:

Definition 1. 

Given the mapping $W:{\left[0,1\right]}^m\to (0,1{]}^m$, it is said to satisfy:

(1)

Normalization: $\sum _{j=1}^m{W}_j\left(X\right)=1$

(2)

Monotonicity: for each $j\in \{1,2,\dots m\}$, there exist $T_j,U_j\in \left(0,1\right)$, and $T_j\le U_j$, such that the vector $W_j\left(X\right)$ is monotonically decreasing with respect to $x_j$ in $[0,T_j]$ and monotonically increasing in $[U_j,1]$;

Then, the vector $W\left(X\right)$ is referred to as an m-dimensional local variable weight vector.

Definition 2. 

Given a mapping $S: {\left[0,1\right]}^m\to {(0,+\infty ]}^m$, if for every $j\in \{1,2,\dots m\}$, there exist $T_j,U_j\in \left(0,1\right)$, and $T_j\le U_j$, it is said to satisfy:

(1)

$W_j\left(X\right)=\frac{W_i^0{S}_i\left(X\right)}{\sum _{k=1}^m{W}_k^0{S}_k\left(X\right)}$ is monotonically decreasing for ${\mathrm{x}}_j$ within $[0,T_j]$, and monotonically increasing within $[U_j,1]$.

(2)

$x_k\le x_i\le T_k\wedge T_i\Rightarrow {\mathrm{S}}_k\left(X\right)\ge {\mathrm{S}}_i\left(X\right),U_k\vee U_i\le x_k\le x_i\Rightarrow {\mathrm{S}}_k\left(X\right)\le {\mathrm{S}}_i\left(X\right)$;

The vector $S\left(X\right)=\left(\begin{array}{c}{S}_1\left(X\right),S_2\left(X\right),\dots ,S_m\left(X\right)\end{array}\right)$ is then referred to as an m-dimensional local state variable weighting vector.

Many scholars have applied the variable-weight theory to the field of bridge assessment [71,72]. The variable-weight evaluation process is as follows:

$$B_i=\sum _{j=1}^m{w}_j\left(x_1,x_2,\dots ,x_m,w_1^{\left(0\right)},w_2^{\left(0\right)},\dots ,w_m^{\left(0\right)}\right)x_j$$

(9)

$$B\left(x_1,\dots ,x_m\right)=\sum _{j=1}^m{x}_j^{\alpha }$$

(10)

$$w_j\left(x_1,\dots ,x_m\right)=\frac{w_j^{\left(0\right)}{x}_j^{\alpha -1}}{\sum _{j=1}^m{w}_j^{\left(0\right)}{x}_j^{\alpha -1}}$$

(11)

$$B_i=\sum _{j=1}^m\frac{w_j^{\left(0\right)}{x}_j^{\alpha }}{\sum _{j=1}^m{w}_j^{\left(0\right)}{x}_j^{\alpha -1}}$$

(12)

In the formula above,

$B_i$—weighted comprehensive evaluation value;

$B(x_1,\dots ,x_m)$—equilibrium function;

$\alpha$—equilibrium parameter;

$w_j^{\left(0\right)}$—the initial weight of the jth indicator and satisfies $\sum _{j=1}^m{w}_j^{\left(0\right)}=1$;

$x_j$—percentage value of the evaluation for the jth indicator;

$w_j(x_1,\dots ,x_m)$—refined weight.

From the above algorithms, it can be seen that the value of the equilibrium parameter $\alpha$ is crucial in variable-weighting calculations. The selection of $\alpha$ will change the overall evaluation results to varying degrees. When $\alpha =1$, corresponding to $S\left(X\right)=(1,...,1)$, it represents a constant-weight model. The transition of $\alpha$ from 1 to 0 reflects the change from a weak to a strong equilibrium. Thus, the smaller the $\alpha$, the greater the impact of local damage on the overall technical condition of the structure.

4. Results and Discussions

4.1. BCI Calculation with Constant Weights

In this study, a specific unit of a viaduct in an actual project was analyzed. Its damage data are listed in

Table 7. Specific damage data for the viaduct unit.

Structural Parts	Viaduct Components	Damage Type	Number of Defects	Unit
Ancillary facilities	Sound barrier	Screen panel damage	1	EA
		Glass shattering	5	EA
	Anti-glare screen	Loose or misaligned	2	EA
	Anti-throwing net (Forbidden grid)	Corrosion of steel structure	0.50	m2
	Gantry (TF pole)	Damage to the height restriction pole	1	EA
		Corrosion of steel structure	0.50	m2
Bridge deck system	Bridge deck pavement	Pothole	0.40	m2
Superstructure	Main girder	Localized concrete breakage	0.01	m2
		Coating flaking	0.05	m2
Substructure	Pier body	Coating flaking	0.10	m2

. Deduction points were determined in accordance with the current standard [45] and Table A2 (Appendix A). The results of the assessment are presented in

Table 8. BCI calculation for the viaduct unit (hundred-mark system).

Structural Parts	Constant-Weight Model in This Study	Model of Current Standard
Ancillary facilities	53.64	/
Bridge deck system	74.21	76.55
Superstructure	95.37	95.37
Substructure	97.63	97.63
Whole viaduct unit	88.71	93.56

.

Compared to the results from the current standard, the BCI score for the whole unit and the deck system obtained by the model proposed in this study are slightly lower, while the superstructure and substructure scores remain unchanged. In addition, ancillary facilities were treated as a separate category in the calculation of the unit BCI, resulting in a score of 53.64. These results indicate that the sufficient evaluation of ancillary facilities has a non-negligible impact on the overall evaluation of the viaduct, highlighting the need for improvement over the current standard. It also validates the rationality of treating ancillary facilities as a separate category.

4.2. Analysis of Equilibrium Parameters in Variable-Weight Models

The evaluation model proposed above has highlighted the impact of ancillary facility damage on the overall technical condition of the viaduct. However, due to their role as expressways or main roads in urban areas and the need for smooth traffic operation, viaducts require special attention to the ancillary facility damage that can result in frequent maintenance or operational safety risks. In situations with high management standards, employing a variable-weight model can further adjust the weights of the viaduct evaluation model, enhancing its sensitivity to localized damage of components. Tuning the equilibrium parameter $\alpha$ (Equation (4)) enables this adjustment. However, it is important to note that higher sensitivity does not always yield better results, as excessive sensitivity of the structural score to component damage can result in wasteful utilization of maintenance resources. Thus, in practical evaluations of viaducts, the determination of a reasonable equilibrium parameter $\alpha$ should be based on management objectives.

This paper presents a case study on the technical condition evaluation of a viaduct to discuss the reasonable value of the equilibrium parameter $\alpha$.

Table 9. Initial BCI scores of the viaduct case (hundred-mark system).

Structural Parts	BCI Score
Ancillary facilities	80.0
Bridge deck system	85.0
Superstructure	90.0
Substructure	95.0
Whole viaduct unit	90.2

displays the initial technical condition scores for the viaduct case. Assuming that each part of the viaduct has different degrees of damage, the corresponding BCI score of the part is 1, 20, 40, 60, or 85, respectively, while at the same time the scores of other parts remain unchanged. For each BCI score assigned to a part, the variation of the whole viaduct score was calculated when the equilibrium parameter $\alpha$ was taken: 0, 0.2, 0.5, 0.8, or 1.0, respectively. The results are shown in Figure 4.

The bridge deck system and ancillary facilities have smaller weights compared to the superstructure and substructure (refer to Table 4). Figure 4 shows that when the damage is slight, the value of the equilibrium parameter $\alpha$ has a minimal impact on the overall structural score for all bridge parts, allowing for evaluation using the constant-weight model ($\alpha =1$). However, in the case of severe damage, the parts with smaller weights and those with larger weights exhibit distinct patterns.

When the damage is severe (e.g., $\mathrm{B}\mathrm{C}\mathrm{I}=10$) as to the components with smaller weights, if $\alpha =0.2$, the overall score falls below 66, as illustrated in Figure 4a,b. This indicates that the viaduct is “failed” according to the current standard [45], aligning with real-world bridge management practices. However, for the parts with larger weights, the overall score remains below 66, even when $\alpha =1$, as depicted in Figure 4c,d. In other words, the parts with larger weights, such as the superstructure and substructure, are highly susceptible to damage, and the constant-weight model adequately fulfills the management requirements. However, the parts with smaller weights, such as the bridge deck system and ancillary facilities, demonstrate lower sensitivity to damage. In cases of severe damage to these parts, it is advisable to assign $\alpha =0.2$ to amplify their influence on the overall viaduct score calculation.

4.3. BCI Calculation with Variable Weights

In cases of higher levels of damage to the bridge deck system and ancillary facilities, the equilibrium parameter $\alpha$ can be set to 0.2. For instance, in Table 8, the ratings for ancillary facilities and the bridge deck system are 53.64 and 74.21, respectively, both indicating a relatively low condition level, suggesting significant damage. Therefore, the equalization parameter is set to 0.2 and the weights are recalculated. Taking ancillary facilities as an example, its weight can be obtained by Equation (11).

$$w_1=\frac{w_1^{\left(0\right)}{x}_1^{0.2-1}}{\sum _{j=1}^4{w}_j^{\left(0\right)}{x}_j^{0.2-1}}=\frac{0.1\times {53.64}^{-0.8}}{0.1\times {53.64}^{-0.8}+0.16\times {74.21}^{-0.8}+0.34\times {95.37}^{-0.8}+0.4\times {97.63}^{-0.8}}=0.15$$

Similarly, the weights of the other parts can be obtained in turn, and then the BCI of the viaduct can be calculated by Equation (12). The results are displayed in

Table 10. Calculation results of the BCI for the technical condition index of the elevated bridge (hundred-mark system).

Structural Parts	BCI (Constant Weights)	Constant Weights	BCI (Variable Weights)	Variable Weights
Ancillary facilities	53.64	0.10	53.64	0.15
Bridge deck system	74.21	0.16	74.21	0.18
Superstructure	95.37	0.34	95.37	0.31
Substructure	97.63	0.40	97.63	0.36
Full bridge	88.71	1.00	86.29	1.00

.

The overall BCI results calculated with variable weights are slightly lower than those with constant weights. After applying variable weights, the weights of the ancillary facilities and bridge deck system increase, while the weights of the superstructure and substructure decrease. These results indicate that the variable-weight evaluation method emphasizes the significance of components with lower weights, under severe damage conditions, to the overall viaduct condition. In such cases, utilizing the variable-weight model is more justified than employing the constant-weight model.

5. Conclusions

There is a lack of technical condition evaluation specifications for viaducts in China currently. An evaluation model for viaducts was proposed in this study based on the AHP, GDM, and Delphi methods. The AHP was adopted to introduce a new indicator layer for ancillary facilities. The weights and deduction points of each layer of the model, as well as the levels of damage to the new components, were determined using the GDM and Delphi methods. In addition, to account for the impact of significant damage to low-weight components on the structural condition, the variable-weight method was adopted to establish a comprehensive evaluation model with variable weights. Finally, the model was verified by utilizing an actual viaduct case. The results indicate that the variable-weight model provides a more accurate representation of the technical condition of viaducts, especially when components are severely damaged. Furthermore, this study examines the conditions suitable for implementing the constant-weight evaluation model and the variable-weight evaluation model, demonstrating that the variable-weight model is recommended when there is a significant disparity in the scores among the viaduct components. The work presented in this paper can support enhanced viaduct evaluation and informed maintenance decision-making, ensuring that the viaducts remain in good operating condition and contribute to urban resilience.

As subsequent operational data accumulates, the evaluation model and weight allocation scheme proposed in this paper can be further refined. For instance, this study employs AHP to assign weights to each component. Despite being adjusted using the GDM method, the outcomes remain subjective, to a certain degree. Subsequent research should focus on refining the weight determination methodology. Ultimately, as constrained by data collection conditions, the limited quantity of viaduct unit spans utilized for validation in this study is inadequate, warranting the inclusion of additional damage examples in future validation endeavors.